Mass transfer steady-state solutions

Steady-state solutions for diffusion/convection When the air is moving, it becomes more difficult to calculate the interception rate. The mass transfer under these circumstances is generally expressed in dimensionless terms. Adam and Delbriick (1968) were able to generate a formula by making the simplifying assumption that the velocity of the air as it passed around the hair was everywhere constant (U) and very similar to the ambient air flow farther away (U0) ... 

Steady state mass or heat transfer in solids and current distribution in electrochemical systems involve solving elliptic partial differential equations. The method of lines has not been used for elliptic partial differential equations to our knowledge. Schiesser and Silebi (1997)[1] added a time derivative to the steady state elliptic partial differential equation and applied finite differences in both x and y directions and then arrived at the steady state solution by waiting for the process to reach steady state. [2] When finite differences are applied only in the x direction, we arrive at a system of second order ordinary differential equations in y. Unfortunately, this is a coupled system of boundary value problems in y (boundary conditions defined at y = 0 and y = 1) and, hence, initial value problem solvers cannot be used to solve these boundary value problems directly. In this chapter, we introduce two methods to solve this system of boundary value problems. Both linear and nonlinear elliptic partial differential equations will be discussed in this chapter. We will present semianalytical solutions for linear elliptic partial differential equations and numerical solutions for nonlinear elliptic partial differential equations based on method of lines. 

When different electrolytes are present on either side of the boundary, the electrolyte distribution is time dependent. This means that an exact thermodynamic solution to the problem is not possible. The solution to the problem given here is a steady-state solution, that is, the solution appropriate to a system in which mass transfer is occurring but under conditions that the liquid junction potential is independent of time. The porous diaphragms described earlier are examples of junctions which meet this condition. There are two well-known solutions to equation (9.7.17), one by Planck [6] and the other by P. Henderson [7]. The latter solution is more often used in practice and therefore is presented here. 

A true steady state can be attained if, for example, the system is confined in a reaction vessel where a solution of A is continuously added to the system while some of the product is continuously removed at the same volume flow rate. Such steady states are by no means exceptional and occur often in living cells or chemical reactors. A steady state then lasts as long as the reaction conditions, including rates of inflow of reactant(s) and outflow of product(s), are kept constant. Also for other rate processes, e.g. involving mass or heat transfer, steady states are often achieved. 

Multiphase reactors can be batch, fed batch, or continuous. Most of the design equations derived in this chapter are general and apply to any of these operating modes. They will be derived for unsteady operation. The unsteady material balances include the inventories in both phases and mass transfer between the phases so that steady-state solutions fonnd by the method of false transients will be true transients if the initial conditions are correct. Compare Section 10.6. 

The simplest treatments of convective systems are based on a diffusion layer approach. In this model, it is assumed that convection maintains the concentrations of all species uniform and equal to the bulk values beyond a certain distance from the electrode, 8. Within the layer 0 x < 5, no solution movement occurs, and mass transfer takes place by diffusion. Thus, the convection problem is converted to a diffusional one, in which the adjustable parameter 8 is introduced. This is basically the approach that was used in Chapter 1 to deal with the steady-state mass transport problem. However, it does not yield equations that show how currents are related to flow rates, rotation rates, solution viscosity, and electrode dimensions. Nor can it be employed for dual-electrode techniques or for predicting relative mass-transfer rates of different substances. A more rigorous approach begins with the convective-diffusion equation and the velocity profiles in the solution. They are solved either analytically or, more frequently, numerically. In most cases, only the steady-state solution is desired. 

In tray columns the first mechanism is dominant. This can lead to a large number of different steady state solutions for a given set of operating conditions. If N is the number of steady states (typically an odd number). Then (N + l)/2 of these steady states are stable. This can lead to complex multi-stable dynamic behavior during column startup and set-point or load changes. These phenomena were observed for vanishing as well as for finite intra-particle mass transfer resistance. An example with a total number of six trays (two reactive and two non-reactive trays plus reboiler and condenser) is shown in Fig. 10.16 for the well-known MTBE process. In contrast to the previous section, the column is now operated in the kinetic... 

For this case, the steady state solution becomes 0=Co for R mass transfer coefficient in the same manner as in the case of diffusion from a single particle. The transient concentration distribution around the particle can be obtained and used to define the mass transfer coefficient for a multiparticle system. 

The goal of DFT modeling is to understand the chain of elementary reaction events in the electrochemical conversion and to calculate the rate constants for these steps. The reaction mechanism and the rate constants, obtained from DFT, are then used to establish and parameterize time-dependent mass balance equations for the adsorbed/desorbed species. The steady-state solution of the surface coverage equations provides the conversion function, which can be used in the simplified current conservation equation in the CL model. The solution of the CL performance model yields the CL polarization curve, which can be used in the fuel cell or stack model. The chain of information transfer looks schematically like... 

Diffusion-Type Mass Transfer Models for Type 1 FacUitation. The state-of-the-art model for Type 1 facilitation is the advancing front model (2,7,8), In this model, the solute is assumed to react instantaneously and irreversibly with the internal reagent at a reaction surface which advances into the globule as the reagent is consumed. A perturbation solution to the resulting nonlinear equations is obtained. In general, the zero-order or pseudo-steady-state solution alone often gives an adequate representation of the diffusion process. 

In a gas absorption process, the solute gas A diffuses into a solvent liquid with which it reacts. The mass transfer is one of steady state unidirectional molecular diffusion and the concentration of A is always sufficiently small for bulk flow to be negligible. Under these conditions the reaction is first order with respect to the solute A. 

A solute diffuses from a liquid surface at which its molar concentration is C, into a liquid with which it reads. The mass transfer rate is given by Fick s law and the reaction is first order with respect to the solute, fn a steady-state process the diffusion rate falls at a depth L to one half the value at the interface. Obtain an expression for the concentration C of solute at a depth z from the surface in terms of the molecular diffusivity D and the reaction rate constant k. What is the molar flux at the surface ... 

Inside a pit in electrolytic solution, anodic dissolution (the critical dissolution current density, and diffusion of dissolved metal hydrates to the bulk solution outside the pit take place simultaneously, so that the mass transfer is kept in a steady state. According to the theory of mass transport at an electrode surface for anodic dissolution of a metal electrode,32 the total increase of the hydrates inside a pit, AC(0) = AZC,<0),is given by the following equation33,34 ... 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

This section concerns the modelling of countercurrent flow, differential mass transfer applications, for both steady-state and non-steady-state design or simulation purposes. For simplicity, the treatment is restricted to the case of a single solute, transferring between two inert phases, as in the standard treatments of liquid-liquid extraction or gas absorption column design. 

The reaction plane model with heterogeneous reactions was discussed at length for acid-base reactions in the previous section. The same modeling technique, of confining the reactions to planes, can be applied to micelle-facilitated dissolution. As with the acid-base model, one starts with a one-dimensional steady-state equation for mass transfer that includes diffusion, convection, and reaction. This equation is then applied to the individual species i, i.e., the solute, s, the micelle, m, and the drug-loaded micelle, sm, to yield... 

Transition Times (sec) to Steady-State Mass Transfer in Laminar Free and Forced Convection along a Planar Electrode, for a Solution of 0.05 M CuS04, 1.5 M H2S04 at 25°C ... 

---